Did you try boiling water to dissolve them?
Also I recommend doing your science experiments in containers that are of little / no value... don't use the expensive cookware.
What was the type of non stick cookware used? If it was teflon, I can't imagine the borax reacting with it so it should dissolve with water, low pH and heat in that order.
If it was a different type of non-stick cookware, it sounds like the borax may have complexed with the metals in the cookware forming an insoluble compound. If this was the case, the cookware is probably permanently damaged and the best you could do is grind the material off which would ruin the surface of the cookware but remove the borate complex.
I don't have a good home remedy, but base will be much more effective than acid in removing/dissolving borax crystals. In a lab/industrial setting I would would use an aqueous NaOH soln. I am not sure if an ammonia based solution will work as well. Soaking at least overnight.